Gregory Chaitin

Gregory John Chaitin (/ˈtʃaɪtɪn/ CHY-tin; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem.[2] He is considered to be one of the founders of what is today known as algorithmic (Solomonoff-Kolmogorov-Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic.[3][4] It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy.

Gregory Chaitin
Born (1947-06-25) 25 June 1947
Chicago[1]
NationalityArgentine-American
Known forChaitin–Kolmogorov complexity
Chaitin's constant
Chaitin's algorithm
Scientific career
FieldsBiology
Mathematics
Computer science
InstitutionsFederal University of Rio de Janeiro
IBM Thomas J. Watson Research Center
InfluencesGottfried Wilhelm Leibniz

Mathematics and computer science

He attended the Bronx High School of Science and City College of New York, where he (still in his teens) developed the theory that led to his independent discovery of algorithmic complexity.[5][6]

Chaitin has defined Chaitin's constant Ω, a real number whose digits are equidistributed and which is sometimes informally described as an expression of the probability that a random program will halt. Ω has the mathematical property that it is definable, with asymptotic approximations from below (but not from above), but not computable.

Chaitin is also the originator of using graph coloring to do register allocation in compiling, a process known as Chaitin's algorithm.[7]

He was formerly a researcher at IBM's Thomas J. Watson Research Center in New York and remains an emeritus researcher. He has written more than 10 books that have been translated to about 15 languages. He is today interested in questions of metabiology and information-theoretic formalizations of the theory of evolution.

Other scholarly contributions

Chaitin also writes about philosophy, especially metaphysics and philosophy of mathematics (particularly about epistemological matters in mathematics). In metaphysics, Chaitin claims that algorithmic information theory is the key to solving problems in the field of biology (obtaining a formal definition of 'life', its origin and evolution) and neuroscience (the problem of consciousness and the study of the mind).

In recent writings, he defends a position known as digital philosophy. In the epistemology of mathematics, he claims that his findings in mathematical logic and algorithmic information theory show there are "mathematical facts that are true for no reason, that are true by accident".[8] Chaitin proposes that mathematicians must abandon any hope of proving those mathematical facts and adopt a quasi-empirical methodology.

Honors

In 1995 he was given the degree of doctor of science honoris causa by the University of Maine. In 2002 he was given the title of honorary professor by the University of Buenos Aires in Argentina, where his parents were born and where Chaitin spent part of his youth. In 2007 he was given a Leibniz Medal[9] by Wolfram Research. In 2009 he was given the degree of doctor of philosophy honoris causa by the National University of Córdoba. He was formerly a researcher at IBM's Thomas J. Watson Research Center and is now a professor at the Federal University of Rio de Janeiro.

Criticism

Some philosophers and logicians disagree with the philosophical conclusions that Chaitin has drawn from his theorems related to what Chaitin thinks is a kind of fundamental arithmetic randomness.[10] The logician Torkel Franzén criticized Chaitin's interpretation of Gödel's incompleteness theorem and the alleged explanation for it that Chaitin's work represents.[11]

Martin Davis (mathematician) says the following about Chaitin's work:

1. Omega is not a constant. It depends on a lot of factors: how programs are represented, how they are generated.

2. There is no "probability of a program halting" because (a) there is no evenly distributed random natural number (program) and (b) because, for example, a program can indefinitely oscillate between halting 1/3 of the time for all inputs less than a constant and 2/3 of the time for all inputs less than a larger constant. It doesn't converge.[12]

3. It makes no sense to say you "improved Godel's Theorem" because Godel's Theorem was superseded by Rosser's more general 1936 extension. Chaitin never mentions Rosser because it can't be explained in simple terms like "this is unprovable" being true and unprovable.

4. For years his definition of Omega the probability of a random program halting summed to more than 1. He says that a program of length N has 2^-N probability of being chosen, but that sums to one for each length, not the sum of all lengths. When he realized that very obvious fact, he added a kludge: if string A is a program and B is a string, then AB is not a program. But that prohibits every programming language including Turing Machines. They all consist of a list of primitives. Add another primitive after any program and that is a program.

5. To try to justify the ambiguity and arbitrariness of Omega, he wrote the Invariance Theorem that the difference between the shortest program to output a given constant in any 2 PLs (programming languages) is bounded. But consider 2 PLs, one of which uses repeated characters for error detection/correction over unreliable communications e.g. the internet. The same programs in the 2 PLs differ in size by the size of the smaller one, which is unbounded. If you follow through his proof you see he makes the mistake of assuming that constants have the same length in both PLs, which is actually similar to what he is trying to prove.

6. The idea that the Godel Sentence, created in the proof of Godel's Theorem, is true "randomly" so it has no proof, misses a fundamental point. We can say it is true only because we can prove it, like any other sentence. However, it is not provable in systems such as Peano Arithmetic. As Godel states, “From the remark that [the unprovable statement] asserts its own unprovability, it follows at once that [the unprovable statement] is correct, since [the unprovable statement] is certainly unprovable (because undecidable). So the proposition which is undecidable in the system PM yet turns out to be decided by meta-mathematical considerations.” [12]

Bibliography

References
  1. Gregory Chaitin (2007), Algorithmic information theory: "Chaitin Research Timeline" Archived 23 March 2012 at the Wayback Machine
  2. Review of Meta Math!: The Quest for Omega, By Gregory Chaitin SIAM News, Volume 39, Number 1, January/February 2006
  3. Calude, C.S. (2002). Information and Randomness: An Algorithmic Perspective. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag.
  4. R. Downey, and D. Hirschfeldt (2010), Algorithmic Randomness and Complexity, Springer-Verlag.
  5. Li; Vitanyi (1997), An Introduction to Kolmogorov Complexity and Its Applications, Springer, p. 92, ISBN 9780387948683, G.J.Chaitin had finished the Bronx High School of Science, and was an 18-year-old undergraduate student at City College of the City University of New York, when he submitted two papers.... In his [second] paper, Chaitin puts forward the notion of Kolmogorov complexity....
  6. Chaitin, G. J. (October 1966), "On the Length of Programs for Computing Finite Binary Sequences", Journal of the ACM, 13 (4): 547–569, doi:10.1145/321356.321363, S2CID 207698337
  7. G.J. Chaitin, Register Allocation and Spilling via Graph Coloring, US Patent 4,571,678 (1986) [cited from Register Allocation on the Intel® Itanium® Architecture, p.155]
  8. Chaitin, G. J. (2003). "From Philosophy to Program Size". arXiv:math/0303352.
  9. Zenil, Hector "Leibniz medallion comes to life after 300 years" Anima Ex Machina, The Blog of Hector Zenil, 3 November 2007.
  10. Panu Raatikainen, "Exploring Randomness and The Unknowable" Notices of the American Mathematical Society Book Review October 2001.
  11. Franzén, Torkel (2005), Gödel's Theorem: An Incomplete Guide to its Use and Abuse, Wellesley, Massachusetts: A K Peters, Ltd., ISBN 978-1-56881-238-0
  12. cite?

12. https://math.stackexchange.com/questions/1491061/paradoxhow-can-godel-prove-that-godel-sentence-is-unprovable-but-true-if-such

Further reading
  • Pagallo, Ugo (2005), Introduzione alla filosofia digitale. Da Leibniz a Chaitin [Introduction to Digital Philosophy: From Leibniz to Chaitin] (in Italian), G. Giappichelli Editore, ISBN 978-88-348-5635-2, archived from the original on 22 July 2011, retrieved 16 April 2008
  • Calude, Cristian S., ed. (2007), Randomness and Complexity. From Leibniz to Chaitin, World Scientific, ISBN 978-981-277-082-0
  • Wuppuluri, Shyam; Doria, Francisco A., eds. (2020), Unravelling Complexity: The Life and Work of Gregory Chaitin, World Scientific, doi:10.1142/11270, ISBN 978-981-12-0006-9
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